We show that if H is an effectively completely decomposable computable torsion-free abelian group, then there is a computable copy G of H such that G has computable orders but not orders of every degree.

We show, for each computable ordinal α and degree $\alpha > {0^{\left( \alpha \right)}}$, the existence of a torsion-free abelian group with proper α th jump degree α.

We extend the notion of limitwise monotonic functions to include arbitrary computable domains. We then study which sets and degrees are support increasing limitwise monotonic on various computable domains. As applications, we provide a characterization of the sets S with computable increasing η-representations using support increasing limitwise monotonic sets on ℚ and note relationships between the class of order-computable sets and the class of support increasing limitwise monotonic sets on certain domains.

It is proved that for every countable structure and a computable successor ordinal α there is a countable structure which is ‐least among all countable structures such that is Σ‐definable in the αth jump. We also show that this result does not hold for the limit ordinal. Moreover, we prove that there is no countable structure with the degree spectrum for.

We study completely decomposable torsion-free abelian groups of the form $\mathcal{G}_S := \oplus_{n \in S} \mathbb{Q}_{p_n}$ for sets $S \subseteq \omega$. We show that $\mathcal{G}_S$has a decidable copy if and only if S is $\Sigma^0_2$and has a computable copy if and only if S is $\Sigma^0_3$.

The main result is that for sets ${S \subseteq \omega + 1}$ , the following are equivalent: The shuffle sum σ(S) is computable.The set S is a limit infimum set, i.e., there is a total computable function g(x, t) such that ${f(x) = \lim inf_t g(x, t)}$ enumerates S.The set S is a limitwise monotonic set relative to 0′, i.e., there is a total 0′-computable function ${\tilde{g}(x, t)}$ satisfying ${\tilde{g}(x, t) \leq \tilde{g}(x, t+1)}$ such that ${{\tilde{f}(x) = \lim_t \tilde{g}(x, t)}}$ (...) enumerates S.Other results discuss the relationship between these sets and the ${\Sigma^0_3}$ sets. (shrink)

We study the complexity of (finitely-valued and transfinitely-valued) Euclidean functions for computable Euclidean domains. We examine both the complexity of the minimal Euclidean function and any Euclidean function. Additionally, we draw some conclusions about the proof-theoretical strength of minimal Euclidean functions in terms of reverse mathematics.

We study what the existence of a classical embedding between computable structures implies about the existence of computable embeddings. In particular, we consider the effect of fixing and varying the computable presentations of the computable structures.